Question

A bullet with mass \(35.5 \mathrm{~g}\) is shot horizontally from a gun. The bullet embeds in a \(5.90-\mathrm{kg}\) block of wood that is suspended by strings. The combined mass swings upward, gaining a height of \(12.85 \mathrm{~cm} .\) What was the speed of the bullet as it left the gun? (Air resistance can be ignored here).

Short answer

Answer: To find the initial speed of the bullet, follow these steps: 1. Identify conservation of momentum in the system. 2. Calculate the final momentum of the system. 3. Identify conservation of energy in the system. 4. Calculate the gravitational potential energy at the highest point of swing. 5. Calculate the initial kinetic energy of the bullet-block system. 6. Solve for the velocity of the bullet-block system after the collision using energy conservation. 7. Calculate the initial velocity of the bullet using momentum conservation. By following these steps and plugging in the given values, you can calculate the initial speed of the bullet as it left the gun.

Step-by-step solution

Identify conservation of momentum

Before the collision, the block is at rest and the bullet is moving. After the collision, both the bullet and the block move together, having the same velocity. In this scenario, the system's momentum is conserved. We will first find the velocity of the bullet-block system just after the collision.

Calculate the final momentum of the system

Using conservation of momentum, we know that the initial momentum of the system equals the final momentum of the system. Let \(v_b\) be the initial velocity of the bullet, and \(v\) be the velocity of the bullet-block system just after the collision. Then, we have: $$(m_b+m_o) v = m_b v_b$$ Where \(m_b\) is the mass of the bullet (0.0355 kg) and \(m_o\) is the mass of the block (5.90 kg), both in SI units.

Identify conservation of energy

Now, let's look at the energy transfer in this problem. The kinetic energy of the bullet-block system just after the collision will be converted into gravitational potential energy when the system reaches the highest point of swing.

Calculate the gravitational potential energy

At the highest point of swing, all kinetic energy is converted into gravitational potential energy. We can calculate the gravitational potential energy using the formula: $$PE = mgh$$ Where \(m\) is the total mass of the block and bullet, \(g\) is the acceleration due to gravity (\(9.81 \mathrm{~m/s^2}\)), and \(h\) is the height gained by the system (0.1285 m).

Calculate the initial kinetic energy

Since energy is conserved, the initial kinetic energy of the bullet-block system can be equated to the gravitational potential energy at the highest point. The kinetic energy can be given as: $$KE = \frac{1}{2}(m_b+m_o)v^2$$

Solve for the velocity of the bullet-block system after the collision

Equating the initial kinetic energy to gravitational potential energy, we can solve for the velocity \(v\) after the collision: $$\frac{1}{2}(m_b+m_o)v^2 = mgh$$ Solve for \(v\) to obtain the velocity of the bullet-block system just after the collision.

Calculate the initial velocity of the bullet

So, we have the expression for velocity after collision, and we know the momentum conservation relationship discussed in Step 2. Now, we can use those relationships to calculate \(v_b\). $$v_b = \frac{(m_b+m_o) v}{m_b}$$ Plug in the known values and calculate the initial speed of the bullet as it left the gun.

Key concepts

Conservation of Energy

The principle of conservation of energy dictates that the total energy in a closed system remains constant over time. This means energy can neither be created nor destroyed but can only change forms. In our bullet-block scenario, the bullet's kinetic energy just before impact is subsequently converted to gravitational potential energy when the system comes to a rest at the highest point of its swing.

By understanding this energy transformation, we can equate the kinetic energy of the system immediately after the collision to the gravitational potential energy at the peak of its rise, allowing us to solve for variables like speed and height.

Kinetic Energy

Kinetic energy is the energy of motion. It is calculated using the formula: \[KE = \frac{1}{2}mv^2\] where \(m\) is the mass of the moving object, and \(v\) is its velocity. The faster an object moves, or the more massive it is, the greater its kinetic energy. In the context of our problem, the kinetic energy of the bullet is crucial as it directly relates to the speed at which the bullet was fired from the gun.

Gravitational Potential Energy

Gravitational potential energy (\(PE\text{_{gravity}}\text{_{height}}\text{_{system}}\)) accounts for the energy an object possesses due to its position in a gravitational field. Calculated with \[PE = mgh\], here \(m\) represents mass, \(g\) is acceleration due to gravity, and \(h\) is the height above the reference point. This form of potential energy is pivotal when objects in the system, like the bullet-block combination, are lifted against gravity's pull.

Momentum Conservation

Momentum conservation is a fundamental concept in physics, expressed as the constant total momentum in a system without external forces. It's given by the product of mass and velocity (\(mv\)) and remains consistent pre- and post-collision. With the bullet and block system, momentum conservation allows us to link the momenta of the two objects, revealing that the initial speed of the bullet equals the combined momentum of the block and bullet after collision.

Inelastic Collisions

An inelastic collision is one where the colliding objects stick together after impact, and kinetic energy is not conserved, though momentum is. These types of collisions are vital to our problem; after the bullet hits the block, they move as a single unit. This change impacts the calculation of final speeds and is a perfect illustration of how inelastic collisions work and how they conserve momentum despite energy being dissipated in other forms.

Mechanical Energy Conservation

The conservation of mechanical energy denotes that, in the absence of nonconservative forces (like friction or air resistance), the sum of kinetic and potential energies in a system remains unchanged. While our bullet-block system makes its pendulum-like swing, assuming no energy lost to the air resistance, the mechanical energy just after the collision (all kinetic) and at the system's highest point (all potential) are equal. This concept allows us to set up equations to find the unknown velocity of the bullet.